me 3230 chapter 9 spatial modeling
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Lecture Slides
ME 3230 Kinematics and Mechatronics
Chapter 9
Spatial Modeling some fundamentals
for Robot Kinematics
By Dr. Debao Zhou
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Scope and Covered Materials
Scope Rotation and homogenous transformation in 3D space
Properties of rotation and homogenous transformation
matrix
Forward kinematics Denavit - HartenbergMethod (D-H Method)
Inverse kinematics
Reading materials: Corresponding materials in Spong and Vidyasagars book,
Chapters 2, 3 and 4.
Sciavicco and Siciliano : Sections 2.1-2.4, 2.7-2.9 and 2.12
Corresponding materials in our Text book
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3D Robot Manipulator
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Robot Hand Position and Orientation Expression
4
P(x, y, z, a b g)
Rigid body - frame/vectorFrame position, orientation
- relative to difference frames
Vector position, orientation
- relative to difference frames
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3D Vector Expressed in Different Frames
Vector rP/O is expressed asp0 in frame
ox0y0z0 and p1 in frame ox1y1z1. The i, j, kare the corresponding unit vectors
1frameto0framefrommatrixrotationtheis10R
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3D Vector Expressed in Different Frames
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Example #1:
Rotation qaround z1
/z0
(positive qfrom frame 0
to frame 1)
Next page: Several examples: i, j, k expressed in 1 to expressed in 0
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Example #1:
Rotation qaround z1
/z0
(positive qfrom 0 to 1)
Meaning? Transform a vector
expression in Frame 1 to frame 0
0
0
1
100
0cossin
0sincos
0
sin
cos
1
1
00 pRp
qq
qq
q
q
0
1
0
100
0cossin
0sincos
0
cos
sin
1
1
00pRp
qq
qq
q
q
Explain q = 90, 180, 270
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Three Simple rotations
Similarly, rotation qaround x0, y0 or z0 axis
qq
qq
q100
0cossin
0sincos
R ,z
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Physical Meaning R01
Frames #0 and #1 share the same origin; R0
1 means the transformation of a vector
expressed in frame #1 to the expression in
frame #0 by using the rotation angle fromframe #0 to #1;
Rmnmeans the transformation of a vector
expressed in frame #n to the expression inframe #m by using the rotation angle from
frame #m to #n.
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Multiple Rotations
p0
= R01
p1p1
= R12 p2
So
p0= R01 R12 p2
And
p0= R0
2 p2
Thus
R02 = R0
1 R12
and R02
(R02
)
T
= 1 11
1 2
3
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Properties of Rotation Matrix
Transposition: Inversion:
Multiple multiplications
Any rotation matrix is Orthogonal Matrices
) ) ) )
) ) ) )
)I
RR
RRRRRRRR
RRRRRRR
RRRRRRRR
n
n
Tn
n
n
n
TTTTn
n
n
n
TTTTn
n
n
n
Tn
n
11
1
3
2
2
1
1
0
1
0
2
1
3
21
1
3
2
2
1
1
0
2
1
3
21
1
3
2
2
1
1
01
3
2
2
1
1
0
)(
)(
)()(
) ) 1000
1
3
2
2
1
1
00
isThere
Set
n
T
n
n
n
n
n
RRR
RRRRR
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Composition of Rotation Matrices
Matrix product is not commutative
Two rotations in general do not commute and itscomposition depends on the order of the single rotations.
Mathematically, AB BA
OrR01 R12R12 R01.
) ) ) )
) ) ) )
)I
RR
RRRRRRRR
RRRRRRR
RRRRRRRR
n
n
Tn
n
n
n
TTTTn
n
n
n
TTTTn
n
n
n
Tn
n
11
1
3
2
2
1
1
0
1
0
2
1
3
21
1
3
2
2
1
1
0
2
1
3
21
1
3
2
2
1
1
01
3
2
2
1
1
0
)(
)(
)()(
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Continues Rotation
Continues rotationaround the current
frame
Body-fixed framerotation
Continues rotation
around a fixed
frame
World-fixed frame
rotation
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Continues Rotation: Euler Angle
Body Fixed Rotation Leonhard Eulerto describe the orientation of a rigid body
in 3D space
Any orientation can be described by three consequence
rotations
15
http://en.wikipedia.org/wiki/Leonhard_Eulerhttp://en.wikipedia.org/wiki/Rigid_bodyhttp://en.wikipedia.org/wiki/Dimensionhttp://en.wikipedia.org/wiki/Dimensionhttp://en.wikipedia.org/wiki/Rigid_bodyhttp://en.wikipedia.org/wiki/Leonhard_Euler -
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Continues Rotation - Euler Angles - YXZ
Rot(Y,1) Rot(X,2) Rot(Z,3)
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Table of Rotation Matrix Euler Anlges
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Homogeneous Transformation
Frames do not share the same origin!
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11
110
11101
11101
11
2
22
0
2
22
1
1
0
2
2
0
0/1
1
1/2
0
1
1
2
0
1
1
11
0
1
1
0
0/1
1
0
0
0
2
22
1
2
2
1
1/2
2
1
1
1
222
2
2
pp
prrRRR
pprRp
pprRp
zyxpT
HHH
H
H
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Homogeneous Transformation
Tzyxp 2222
2
1
1
1
33
1
131
31
0H
H0
H
0H
dRR
d
R
dR
TT
T
TT
2
2p
11p
0
0p
0
0/1r
1
1/2r
P
1
1/2
2
2
2
1
1
1/2
2
2
1
1 rpRrpp
0
0/1
1
1
1
0
0
0/1
1
1
0
0 rpRrpp
0
0/1
1
1/2
2
2
1
2
1
0 rrpRR
00/1
11/2
10
22
21
10 rrRpRR
2
2p
1
1p
0
0p
0
0/1r
1
1/2r
P
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What is the benefit to us?
Robot manipulator express the end-effector
vector to the base frame. Fundamentally
different from those in Chapter 5
2
2p
1
1p
0
0p
0
0/1
r
1
1/2r
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Basic Homogeneous Transformation
Translation
Simple Rotation
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Defining the Homogeneous Transformation Matrix
It is a 4x4 Matrix that describes 3-Space with
information that relates Orientation and Position (pose) ofa remote space to a local space
nx ox ax dxny oy ay dynz oz az dz0 0 0 1
n vectorprojects theXrem Axis to theLocalCoordinate
System
o vectorprojectsthe YremAxis to theLocal
CoordinateSystem
a vectorprojects theZrem Axis tothe LocalCoordinateSystem
d vector isthe positionof the originof theremotespace inLocalCoordinatedimensions
This 3x3 Sub-Matrix isthe information thatrelates orientation of
Framerem to FrameLocal(This is called R therotational Submatrix)
11101
11101
1
11
0
1
1
0
0/1
1
0
0
0
222
1
22
11/2
21
11
pprRp
pprRp
H
H
2
2p
11p
0
0p
0
0/1r
1
1/2r
P
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Defining the Homogeneous Transformation Matrix
nx ox ax dxny oy ay dynz oz az dz0 0 0 1
Perspectiveor ProjectionVector
ScalingFactor
This matrix is a transformationtool for space motion!
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HTM A Physical Interpretation
1. A representation of a coordinate transformation
relating the coordinates of a point P between 2different coordinate systems
2. A representation of the position and orientation(pose) of a transformed coordinate frame in thespace defined by a fixed coordinate frame
3. An operation that takes avectorP and rotates
and/or translates it to anew vectorPtin the samecoordinate frame
2
2p
1
1p
0
0p
0
0/1r
1
1/2r
P
11101
1
11
0
1
1
0
0/1
1
0
0
0 pprRp
H
10
0
0/1
1
01
0
rR
H
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Looking Closely at the T0n Matrix
T0n
matrix related theend of the arm frame
(n) to its base (0)
Thus it must contain
information related tothe several joints of the
robot
It is a 4x4 matrix
populated by complexequations for both
position and orientation
(POSE)
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Looking Closely at the T0n Matrix
To get at the equation set, wewill choose to add a series ofcoordinate frames to each ofthe joint positions
There are n+1 frames n+1 rigid bodies = number offrames Why?
njoints
The frame attached to the 1st jointis labeled 0 the base frame!while joint two has Frame 1, etc.
The last Frame is the end ornFrame
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Looking Closely at the T0n Matrix
As we have seen earlier, wecan define a HTM (T(i-1)i) to the
transformation between any
two consecutive frames
Thus we will find that the T0n isgiven by a product formed by:
n
n
n
TTTT 12
1
1
00
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Looking Closely at the T0n Matrix
For simplicity, we will redefineeach of the of these transforms(T(i-1)
i) as Ai
Then, for a typical 3 DOF robot
Arm, T03 = A1*A2*A3
While for a full functioned 6 DOFrobot (arm and wrist) would be:
T0n = A1*A2*A3*A4*A5*A6
A1 to A3explain the arm jointeffect while A4 to A6 explain thewrist joint effects
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Frame To Frame Arrangements in 3D Space
When we move from one frame to another,we will encounter:
Two translations (in a controlled sense)
Two rotations (also in a controlled sense) A rotation and translation WRT the Zi-1
These are called the Joint Parameters
A rotation and translation WRT the Xi These are called the Link Parameters
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Relationship between Two Frames
Rotate qaround zi-1, translate dalongzi-1, translate a
alongxiaxis, and rotate a aroundxi axis.
Frame (i-1) on link (i-1)
Frame i on link i
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Talking Specifics Joint Parameters
qi is an angle measured about the Zi-1 axisfrom Xi-1 to Xi and is a variable for arevolute joint its fixed for a Prismatic Joint
di is a distance measured from the origin ofFrame(i-1) to the intersection of Zi-1 and Xiand is a variable for a prismatic joint its
fixed for a Revolute Joint
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Talking Specifics Link Parameters
aiis the Link length and measures the distance
from the intersection ofZi-1 to the origin of Frameimeasured alongXi
ai is the Twist angle which measures the angle
from Zi-1 to Zias measured aboutXi Both of these parameters are fixed in valueregardless of the joint type. A Further note: Fixed does not mean zero degrees or
zero lengthjust that they dont change
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Returning to the 4 Frame-Pair Operators:
1st is q which is an
operation ofpure
rotation about Zor:
2nd is d which is a
translation along Zor:
os 0 0
0 0
0 0 1 0
0 0 0 1
C SinSin Cos
q q
q q
1 0 0 0
0 1 0 00 0 1
0 0 0 1
d
O
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Returning to the 4 Frame Operators:
3rd
is a TranslationAlong Xor:
4th is awhich is a
pure Rotation
about Xor:
1 0 0
0 1 0 0
0 0 1 0
0 0 0 1
a
1 0 0 00 0
0 0
0 0 0 1
Cos Sin
Sin Cos
a a
a a
Th O ll Eff t i
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The Overall Effect is:
os 0 0 1 0 0 0 1 0 0 1 0 0 0
0 0 0 1 0 0 0 1 0 0 0 0
0 0 1 0 0 0 1 0 0 1 0 0 0
0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1
C Sin a
Sin Cos Cos Sin
d Sin Cos
q q
q q a a
a a
aq ,,,,1 xaxdzzii
i RotTranTranRotAT
Si lif i thi M t i P d t
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Simplifying this Matrix Product:
0
0 0 0 1
i i i i i i i
i i i i i i i
i i i
C S C S S a C
S C C C S a S S C d
q q a q a q
q q a q a q a a
This matrix is the general transformation relating each
and every of the frame pairs along a robot structure
aq ,,,,1 xaxdzzi
i
i RotTranTranRotAT
M th ti C d
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Mathematica Code
M th ti C d
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Mathematica Code
R b t Ki ti
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Robot Kinematics
Foreword Kinematics Definition: Given the joint variable of the robot ,
to determine the position and orientation of the
end-effector.
Inverse Kinematics
Definition: Given a desired position and
orientation of the end-effector of a robot,
determine a set of joint variables that achievethe desired position and orientation.
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F d Ki ti Di t Ki ti
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Denavit-HartenbergMethod
or D-H Convention A Step-by-Step approach for modeling each of the
frames from the initial (or 0) frame all the way to the n(or end) frame
This modeling techniquemakes each joint axis
(either rotation or translation)
the Z-axis of the appropriate
frame (Z0 to Zn-1). The Joint motion, thus, is taken
W.R.T. the Zi-1 axis of the frame
pairmaking up the specific transformation matrix
under design42
Foreword Kinematics Direct Kinematics
A l i D H t G l C
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Applying D-H to a General Case:
Link i:
Connect frame i -1and frame i
Connect joint iand joint i+1
Think: Link i -1as theground link (or link 0):
- Joint 1 and frame 0
D it H t b R f F L t
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Denavit-Hartenberg Reference Frame Layout
http://www.youtube.com/watch?v=rA9tm0gTln8
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Th D H M d li R l
http://www.youtube.com/watch?v=rA9tm0gTln8http://www.youtube.com/watch?v=rA9tm0gTln8http://www.youtube.com/watch?v=rA9tm0gTln8http://www.youtube.com/watch?v=rA9tm0gTln8 -
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The D-H Modeling Rules:
1) Locate and mark the
motion ( using qiand di)
and label the joint axes:Z0 to Zn-1
2) Establish the Base
Frame. Set Base Origin
anywhere on the Z0axis. ChooseX0and Y0
conveniently and to
form a right hand frame.
0) Identify links and joints (motions)
n+1 links from 0 to n, n joints for n+1 links;
The D H Modeling R les
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The D-H Modeling Rules:
3) Locate the origin Oiwhere the commonnormal to Zi-1 and Ziintersects Zi. IfZi
intersects Zi-1 locate Oiatthis intersection. IfZi-1and Ziare parallel, locateOiat Joint i+1
OrXihas to be perpendicularwith Ziand Zi-1 andconnect with Ziand Zi-1
The intersection points formsthe origin of the frames
n+1 links from 0 to n, njoint from 0 to n-1;
The D H Modeling Rules:
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The D-H Modeling Rules:
4) EstablishXialong the common normalbetween Zi-1 and ZithroughOi, or in the
direction Normal to the plane Zi-1 Zi if
these axes intersect5) Establish Yi to form a right hand system
Set i = i+1, ifi
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The D-H Modeling Rules:
6) Establish the End-Effector (n) frame:OnXnYnZn. Assum ing the n
thjo int is
revolute, set kn = a along the direction Zn-1.
Establish the origin On conveniently alongZn, typically mounting point of gripper or
tool. Setjn= o in the direction of gripper
closure (opening) and set in
= nsuch that n
= oxa. Note if tool is not a simple gripper,
setXn and Yn conveniently to form a right
hand frame.
The D H Modeling Rules:
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The D-H Modeling Rules:
7) Create a table of Link parameters:1) qias angle about Zi-1between Xs
2) dias distance along Zi-1
3) aias angle aboutX
ibetween Zs
4) aias distance alongXi
8) Form HTM matricesA1,A2, An by
substituting q, d, aand a into the general
model
9) Form T0n =A1*A2**An
Some Issues to remember:
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Some Issues to remember:
Xi
is perpendicular with Zi-1
and intersect with Zi-1
.
Xaxis If you have parallel Zaxes, theXaxis of the second
frame runs perpendicularly between them
Kinematic Home When working on a revolute joint, the model will besimpler if the twoXdirections are in alignment atKinematic Home i.e. Our models starting point (q=0)
To achieve this kinematic home, rotate the model aboutmoveable axes (Zi-1s) to alignXs or set q= 0
Kinematic Home is not particularly criticalforprismaticjoints
Some Issues to remember:
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Some Issues to remember:
An ideal model will have n+1 frames However, additional frames may be
necessarythese are considered Dummy
frames since they wont contain joint axes
Three link Planar Arm
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Three-link Planar Arm
Three link Planar Arm
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Three-link Planar Arm
Individual Homogenous TransformationMatrix
Homogenous Transformation Matrix from 0
to 3
T03 = A 1 A 2 A 3 =
2 DOFs (Motions)
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2-DOFs (Motions)
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x0
y0
x1
z1
z2
x2
q
d
q
Example 1: 6 dofs Articulating Arm
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Example 1: 6-dofs Articulating Arm
D H Table
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D-H Table
Frames Link Var q d a a Sa Ca Sq Cq
0 to1 1 R q1 0 0 90 -1 0 S1 C1
1 to2 2 R q2 0 a2 0 0 1 S2 C2
2 to 3 3 R q3 0 a3 0 0 1 S3 C3
3 to 4 4 R q4 0 a4 -90 -1 0 S4 C4
4 to 5 5 R q5 0 0 90 1 0 S5 C5
5 to 6 6 R q6* d6 0 0 0 1 S6 C6
* With End Frame in Better Kinematic Home. Here,
as shown, it would be (a6 - 90), which is a problem!
A Matrices
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A Matrices
1
1 0 1 0
1 0 1 0
0 1 0 0
0 0 0 1
C S
S CA
2
2
2
2 2 0 2
2 2 0 2
0 0 1 0
0 0 0 1
C S a C
S C a S A
3
33
3 3 0 3
3 3 0 30 0 1 0
0 0 0 1
C S a C
S C a S A
4
44
4 0 4 4
4 0 4 40 1 0 0
0 0 0 1
C S a C
S C a S A
A Matrices cont
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A Matrices, cont.
5
5 0 5 0
5 0 5 0
0 1 0 0
0 0 0 1
C S
S CA
6
6
6 6 0 0
6 6 0 0
0 0 1
0 0 0 1
C S
S CA
d
At Better Kinematic Home!
Leads To:
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Leads To:
Forward Kinematics is
After Preprocessing A2-A3-A4,
with (Full) Simplify function, the
FKS must be reordered as A1-
A234-A5-A6 and Simplified
6
5
2
1
1
0
6
0 TTTT
Solving for FKS
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Solving for FKS
Pre-process {A2*A3*A4} with Full Simplify They are the planer arm issue as in the
previous robot model
Then Form: A1* {A2*A3*A4}*A5*A6 Simplify for FKS!
Simplifies to (in the expected Tabular Form):
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Simplifies to (in the expected Tabular Form):
nx = C1(C5C6C234 - S6S234) - S1S5C6
ny = C1S5C6 + S1(C5C6C234 - S6S234)nz = S6C234 + C5C6S234
ox = S1S5S6 - C1(C5S6C234 + C6S234)
oy = - C1S5S6 - S1(C5S6C234 + C6S234)
oz = C6C234 - C5S6S234
ax = C1S5C234 + S1C5
ay = S1S5C234 - C1C5
az = S5S234
dx = C1(C234(d6S5 + a4) + a3C23 + a2C2) + d6S1C5dy = S1(C234(d6S5 + a4) + a3C23 + a2C2) - d6C1C5
dz = S234(d6S5 + a4) + a3S23 + a2S2
Verify the Model
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Verify the Model
Again, substitute knowns (typically 0
or 0units) for the variable kinematic variables
Check each individual A matrix against
your robot frame skeleton sketch for
physical agreement
Check the simplified FKS against the
Frame skeleton for appropriateness
After Substitution:
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After Substitution:
nx = C1(C5C6C234 - S6S234) - S1S5C6 = 1(1-0) 0 = 1
ny = C1S5C6 + S1(C5C6C234 - S6S234) = 0+ 0(1 0) = 0 nz = S6C234 + C5C6S234 = 0 + 0 = 0
ox = S1S5S6 - C1(C5S6C234 + C6S234) = 0 1(0 + 0) = 0
oy = - C1S5S6 - S1(C5S6C234 + C6S234) = -0 0(0 + 0) = 0
oz = C6C234 - C5S6S234 = 1 0 = 1
ax = C1S5C234 + S1C5 = 0 + 0 = 0
ay = S1S5C234 - C1C5 = 0 1 = -1
az = S5S234 = 0
dx = C1(C234(d6S5 + a4) + a3C23 + a2C2) + d6S1C5= 1*(1(0 + a4) + a3 + a2) + 0 = a4 + a3 + a2
dy = S1(C234(d6S5 + a4) + a3C23 + a2C2) - d6C1C5= 0(1(0 + a4) + a3 + a2) d6 = -d6
dz = S234(d6S5 + a4) + a3S23 + a2S2= 0(0 + a4) + 0 + 0 = 0
Inverse Kinematics
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Inverse Kinematics
Based on the direct (forward) kinematicsand the special properties of HTM
properties to calculate the joint parameters
(angle for revolute joint and moving
distance for prismatic joint)